Homogeneous dynamical systems theory

We study controlled homogeneous dynamical systems and derive conditions under which the system is perspective controllable. We also derive conditions under which the system is observable in the presence of a control over the complex base field. In the absence of any control input, we derive a necessary and sufficient condition for observability of a homogeneous dynamical system over the real base field. The observability criterion obtained generalizes a well known Popov-Belevitch-Hautus rank criterion to check the observability of a linear dynamical system. Finally, we introduce rational, exponential, interpolation problems as an important step toward solving the problem of realizing homogeneous dynamical systems with minimum state dimensions.